Liouville type theorems for singular integral equations and integral systems

In this paper, we establish some 
Liouville type theorems for positive solutions of some integral 
equations and integral systems in $R^N$. The main technique we use 
is the method of moving planes in an integral form.


1.
Introduction. In this paper, we study the nonexistence of positive solutions for the following integral equation and integral system |y| t |x − y| N −α g(u(y), v(y)) dy in R N , (2) where N ≥ 2, 0 < α < N and 0 ≤ t < α. The solutions of (1) and (2) are closely related to the following partial differential equation and partial differential system in R N (−∆) and In fact, every positive smooth solution of (1) or (2) multiplied by a constant satisfies (3) or (4) respectively. Here (−∆) α 2 is defined by (−∆) where ∧ is the Fourier transformation and ∨ is its inverse.
In this paper, we are interested in the nonexistence results of (1) and (2). When α = 2 and t = 0, there are a great number of results on the nonexistence results of (3) and (4), see [1,3,4,14,15,16,29]. The first result is [15], in which the authors proved, among other things, that if f (u) = u p , then problem (3) has no positive C 2 solutions provided 0 < p < N +2 N −2 . This result is optimal in the sense that for any p ≥ N +2 N −2 , there are infinitely many positive solutions to (3). Thus the Sobolev exponent N +2 N −2 is the dividing exponent between existence and nonexistence of positive solutions. It is natural to ask whether similar results hold for system of It has been conjectured, see for example [13], that the hyperbola is the dividing curve between existence and nonexistence for problem (5). This conjecture is supported by the results that there are no radial positive solutions to (5) provided that p, q satisfy 1 p+1 + 1 q+1 > N −2 N , see Mitidieri [28] for p > 1, q > 1 and Serrin and Zou [30] for p > 0, q > 0. Moreover, it is proved in [32] that there are indeed infinitely many radial solutions provided that 1 p+1 + 1 q+1 ≤ N −2 N . This solves the conjecture in the sense of radial case. As for the non-radial case, this conjecture has not been solved completely, we refer the readers to [13,28,30,31,33]. We will not give further comments on these results. We note that all these results concerning the particular nonlinear term f (u) = u p . A natural question is whether problem (3) possesses a positive solution for general nonlinearity f (u). As for α = 2, this question has been solved by L.Damascelli and F.Gladiali in [11]. They proved that problem (3) does not possess positive solutions when f is subcritical. The main tool they used is the method of moving planes. We note that f is only required to be continuous in this paper. So we can not conclude that the weak solutions of problem (3) are of C 2 class. Moreover, since f is not assumed to be Lipschitz continuous, the usual maximum principle does not work. In developing the method of moving plane, the authors in [11] used the technique based on integral inequalities, an idea originally due to S.Terracini's work [34] and [35]. After the work of [11], Y.Guo and J.Liu obtained the Liouville type results of the elliptic system, i.e., problem (4) with α = 2 and t = 0 in [17].
We note that all the results listed above are concerning the case α = 2 and t = 0. In a recent paper [36], we established Liouville type theorems for integral equation (1) and integral system (2) with 0 < α < N and t = 0. In this paper, we want to know whether similar results hold for 0 < α < N and 0 ≤ t < α. Up to now, there are a lot of works on this type of integral equations and integral systems. For example, in [23], the authors studied integral system (2) In the same spirit of this, W.Chen and C.Li studied the Lane-Emden conjecture in [5]. Recently, Y.Fang and W.Chen studied the Liouville type theorem for poly-harmonic equation in R N + in [12]. The main ingredient in these papers is the moving plane method based on the maximum principle of integral forms, which was first introduced by W.Chen, C.Li and B.Ou in [9] and [10] and then used widely in [6,8,18,19,20,24,25,26,27]. For more details, please see also [7] for a survey. At the same time, moving spheres based on the maximum principle of integral forms was also introduced by Y.Li in [21]. Inspired by these results, we study the Liouville type results for problems (1) and (2), i.e., the general nonlinear problems. Our first result concerns the Liouville type result of problem (1). Our first result is the following Then u ≡ 0.
Next, we study the integral system (2). We first study a simpler problem that is, we assume f depends only on v and g depends only on u. Our second result is the following be a positive solution of problem (7). Suppose that f, g : [0, +∞) → R are continuous and satisfy (i) f (s), g(s) are nondecreasing in (0, +∞); are nonincreasing in (0, +∞); (iii) either h or k is not a constant. Then (u, v) ≡ (0, 0). Theorem 1.2 can be extended to more general case, in which both f and g depend on (u, v). We have the following be a positive solution of problem (2). Suppose that f, g : [0, +∞) × [0, +∞) → R are continuous functions satisfying (i) f (s, l), g(s, l) are nondecreasing in l for fixed s and f (s, l), g(s, l) are nondecreasing in s for fixed l; (ii) there exist p 1 , q 1 ≥ 0, p 1 + q 1 = N +α−2t N −α such that h(s, l) = f (s,l) s p 1 l q 1 is nonincreasing in l for fixed s and h(s, l) is nonincreasing in s for fixed l; (iii) there exist p 2 , q 2 ≥ 0, p 2 + q 2 = N +α−2t N −α such that k(s, l) = g(s,l) s p 2 l q 2 is nonincreasing in l for fixed s and k(s, l) is nonincreasing in s for fixed l; (iv) either h or k is not a constant. Then (u, v) ≡ (0, 0).
Using the same technique we can consider the more general system We have the similar result as Theorem 1.3. Theorem 1.4. Let (u, v) ∈ C 0 (R N )×C 0 (R N ) be a positive solution of problem (8).
.., n are continuous functions satisfying (i) f (s, l), g(s, l) are nondecreasing in l for fixed s and f (s, l), g(s, l) are nondecreasing in s for fixed l; s p 1i l q 1i is nonincreasing in l for fixed s and h i (s, l) is nonincreasing in s for fixed l; s p 2i l q 2i is nonincreasing in l for fixed s and k i (s, l) is nonincreasing in s for fixed l; (iv) there exist 1 ≤ i, j ≤ n such that either h i or g j is not a constant. Then (u, v) ≡ (0, 0). According to Theorem 1.1 to Theorem 1.4, we have the following corollaries.
This paper is organized as follows. We prove Theorem 1.1 and Theorem 1.2 in Section 2 and Section 3 respectively. Theorem 1.3 and Theorem 1.4 are proved in Section 4. In the following, we denote by C a positive constant which may vary from line to line.

2.
Proof of Theorem 1.1. Let us study the positive solutions to the integral equation Since we don't know the behaviors of u at infinity, we introduce the Kelvin's trans- Since u is continuous in R N , we conclude that v is continuous and strictly positive in R N \ {0} with possible singularity at the origin. Moreover, v decays at infinity as Now we use the moving plane method to prove our result. For a given real number λ > 0, define Since |x − y| = |x λ − y λ | and |x − y λ | = |x λ − y|, then the assertion of this lemma holds easily.
Proof. First we note that for any λ > 0, we have |y| > |y λ | for ∀y ∈ Σ λ . If we denote by Then by Lemma 2.
By the Hardy-Littlewood-Sobolev inequality, see for example [22], it follows that for any Since λ must be measure 0 and hence empty.
We now move the plane x 1 = λ 0 to the left as long as v(x) ≤ v λ (x) holds for ∀x ∈ Σ λ . Suppose that this procedure stops at some λ = λ 1 , then we have v(x) ≤ v λ1 (x). Moreover, we have the following result.
Proof. We prove the conclusion by contradiction. We will prove that if λ 1 > 0, then the plane can be moved further to the left. More precisely, there exists an ε > 0, such that v(x) ≤ v λ (x), x ∈ Σ λ for all λ ∈ [λ 1 − ε, λ 1 ]. This contradicts the choice of λ 1 .
Proof of Theorem 1.1. Let v be the Kelvin's transformation of u at point p ∈ R N , we first prove that v is radially symmetric. We use the method of moving plane and prove the symmetry in every direction. Without loss of generality, we choose the x 1 direction and prove that v is symmetric in the x 1 direction. We can carry out the procedure as the above and conclude that λ 1 ≤ 0. Then we conclude by continuity that v(x) ≤ v 0 (x) for all x ∈ Σ 0 . We can also perform the moving plane procedure from the left and find a corresponding λ 1 ≥ 0. Then we get v 0 (x) ≤ v(x) for x ∈ Σ 0 . This fact and the above inequality imply that v(x) is symmetric with respect to T 0 . We perform the above procedure in every direction, then v and hence u are radially symmetric around the point of the Kevin's transform. Since p is arbitrary, we conclude that u is constant. Finally, we infer from equation (1) that u ≡ 0.
3. Proof of Theorem 1.2. The spirit of the proof of Theorem 1.2 is similar to the proof of Theorem 1.1. We denote by Σ λ , T λ , x λ and u λ (x) as above. We consider the Kelvin's transform w, z of u, v defined by Then a direct calculation implies that w, z satisfy It is easy to see that w, z are continuous and strictly positive in R N \ {0} with possible singularity at the origin. Moreover, they decay at infinity as u(0) 1 |x| N −α and v(0) 1 |x| N −α respectively, so we have w, z ∈ L τ +1 ∩ L ∞ (R N \ B r (0)) for any r > 0 with τ = N +α N −α . In the spirit of Lemma 2.1, we have the following result.
Proof. The proof of this lemma is similar to the proof of Lemma 2.1, we omit it.
We now move the plane x 1 = λ 0 to the left as long as w(x) ≤ w λ (x) and z(x) ≤ z λ (x) hold for ∀x ∈ Σ λ . Suppose that this procedure stops at some λ = λ 1 , then we have w(x) ≤ w λ1 (x) and z(x) ≤ z λ1 (x) for ∀x ∈ Σ λ1 . Moreover, we have the following result. Proof. We prove the conclusion by contradiction. We will prove that if λ 1 > 0, then the plane can be moved further to the left. More precisely, there exists an ε > 0, such that w(x) ≤ w λ (x) and z(x) ≤ z λ (x), x ∈ Σ λ for all λ ∈ [λ 1 − ε, λ 1 ]. Then this contradicts the choice of λ 1 .

Now by equations
= 0 and hence Σ w λ , Σ z λ must be empty. Proof of Theorem 1.2. Let w, z be the Kelvin's transformation of u, v at p, we first prove that w, z are radially symmetric. We still use the method of moving plane and prove the symmetry in every direction. Without loss of generality, we choose the x 1 direction and prove that w, z are symmetric in the x 1 direction. We can carry out the procedure as the above and conclude that λ 1 ≤ 0, then we conclude by continuity that w(x) ≤ w 0 (x) and z(x) ≤ z 0 (x) for all x ∈ Σ 0 . We can also perform the moving plane procedure from the left and find a corresponding λ 1 ≥ 0, that is, w 0 (x) ≤ w(x) and z 0 (x) ≤ z(x) for x ∈ Σ 0 . This fact and the above inequality imply that w(x), z(x) are symmetric with respect to T 0 . We perform the above procedure in all directions and for all p ∈ R N , then w, z and hence u, v are radially symmetric around the point of the Kevin's transform. Since p is arbitrary, we conclude that u, v are constants. Since either h or k is not a constant, it follows that u = v ≡ 0. 4. Proofs of Theorem 1.3 and Theorem 1.4. Now we prove the general results for problem (2), i.e., Theorem 1.3 and Theorem 1.4. The spirit of the proofs is still the moving plane method, but the calculation is more complicated. We only prove Theorem 1.3, the proof of Theorem 4 is similar and we omit it. We still denote by Σ λ , T λ , x λ and u λ (x) as above. We also consider the Kelvin's transform w, z of u, v defined by (2), then a direct calculation implies that w, z satisfy By the definitions of h(s, l) = f (s,l) s p 1 l q 1 , k(s, l) = g(s,l) s p 2 l q 2 , we have where p 1 + q 1 = p 2 + q 2 = N +α−2t N −α . For the same reason we have that w, z are continuous and strictly positive in R N \ {0} with possible singularity at the origin. Moreover, they decay at infinity as u(0) 1 |x| N −α and v(0) 1 |x| N −α respectively, so we have w, z ∈ L τ +1 ∩ L ∞ (R N \ B r (0)) for any r > 0 with τ = N +α N −α . In the spirit of Lemma 2.1 and Lemma 3.1, we have the following result.
By the definitions of h(s, l) and k(s, l), the above two equations can be written as Proof. The proof is a direct calculation, see the proofs of Lemma 2.1 and Lemma 3.1.
Proof of Theorem 1.3. Let w, z be the Kelvin's transformation of u, v at p, we first prove that w, z are radially symmetric. We still use the method of moving plane and prove the symmetry in every direction. Without loss of generality, we choose the x 1 direction and prove that w, z are symmetric in the x 1 direction. We can carry out the procedure as the above and conclude that λ 1 ≤ 0, then we conclude by continuity that w(x) ≤ w 0 (x) and z(x) ≤ z 0 (x) for all x ∈ Σ 0 . We can also perform the moving plane procedure from the left and find a corresponding λ 1 ≥ 0, and then we get w 0 (x) ≤ w(x) and z 0 (x) ≤ z(x) for x ∈ Σ 0 . This fact and the above inequality imply that w(x), z(x) are symmetric with respect to T 0 . we perform the above procedure in every direction, then it follows that w, z and hence u, v are radially symmetric around the point of the Kevin's transform. Since p is arbitrary, we conclude that u, v are constants. By the assumption that either h(t) or k(t) is not a constant, we deduce that u = v ≡ 0.